# Piecewise Functions

## A Function Can be in Pieces

We can create functions that behave differently based on the input (x) value.

A function made up of 3 pieces

### Example:

- when x is less than 2, it gives
**x**^{2}, - when x is exactly 2 it gives
**6** - when x is more than 2 and less than or equal to 6 it gives the line
**10-x**

It looks like this:

(a solid dot means "including",

an open dot means "not including")

And this is how we write it:

The Domain (all the values that can go into the function) is all Real Numbers up to and including 6, which we can write like this:

Dom(f) = (-¡Ä, 6] (using Interval Notation)

Dom(f) = {x | x ¡Â 6} (using Set Builder Notation)

And here are some example values:

### Example: Here is another piecewise function:

| | which looks like: | | |

What is h(-1)? | x is ¡Â 1, so we use h(x) = 2, so **h(-1) = 2** |

What is h(1)? | x is ¡Â 1, so we use h(x) = 2, so **h(1) = 2** |

What is h(4)? | x is > 1, so we use h(x) = x, so **h(4) = 4** |

Piecewise functions let us make functions that do anything we want!

### Example: A Doctor's fee is based on the length of time.

- Up to 6 minutes costs $50
- Over 6 to 15 minutes costs $80
- Over 15 minutes costs $80 plus $5 per minute above 15 minutes

Which we can write like this:

If you were there for 12 minutes what would the fee be? $80

If you were there for 20 minutes what would the fee be? $80+$5(20-15) = $105